1. example: four masses on springs
Normal Modes
example: four masses on springs

2. Four masses on springs
Find a physical description of a system that might look like this:
We will use:
Newton’s laws
Vectors
Matrices
Linearity (superpositions)
Complex numbers
Differential equations
Exponential functions
Eigenvalues
Eigenvectors

3. Problem: masses on springs (I)
We consider four masses connected to springs with spring constant k and their motion restricted to one spatial dimension.
Step 1: Write down Newton’s law for the motion of the masses

4. Problem: masses on springs (I)
Step 2: Combine the degrees of freedom into a vector and write the equations of motion as a matrix equation
Step 3: Use a complex exponential as the ansatz for the solution to this equation

5. Problem: masses on springs (II)
Step 4: Substitute this ansatz into the equation of motion
Step 5: Solve the eigenvalue equation for eigenvalues 2 and eigenvectors v
We do not need the negative frequency solutions since we only consider the real part as physically relevant

6. Problem: masses on springs (III)
Find the eigenvectors (here not normalized) from the corresponding homogeneous equations
Step 6: The general solution is then given by a superposition of all these normal modes with complex amplitudes A1, A2, A3, A4 chosen to meet the initial conditions:
If the system is in one of these normal modes (i.e. all Ai zero except An) all masses will oscillate with the same frequency n=n(k/m)1/2 and constant amplitude ratios defined by vn .

7. Problem: masses on springs (IV)
Visualization of the normal modes